Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Proposition 1.4.3.3. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be categories and let $e: \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ denote the evaluation functor, given on objects by the formula $e(F,C) = F(C)$. Then the composite map

\[ \operatorname{N}_{\bullet }( \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) ) \times \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \simeq \operatorname{N}_{\bullet }( \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \times \operatorname{\mathcal{C}}) \xrightarrow { \operatorname{N}_{\bullet }(e) } \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) \]

corresponds, under the bijection of Proposition 1.4.3.2, to an isomorphism of simplicial sets $\rho : \operatorname{N}_{\bullet }( \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) ) \rightarrow \operatorname{Fun}( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}), \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) )$.

Proof. For each $n \geq 0$, the map $\rho $ is given on $n$-simplices by the composition

\begin{eqnarray*} \operatorname{Hom}_{ \operatorname{Set_{\Delta }}}( \Delta ^ n, \operatorname{N}_{\bullet }( \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) ) ) & \simeq & \operatorname{Hom}_{\operatorname{Cat}}( [n], \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) ) \\ & \simeq & \operatorname{Hom}_{ \operatorname{Cat}}( [n] \times \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \\ & \xrightarrow {v} & \operatorname{Hom}_{ \operatorname{Set_{\Delta }}}( \operatorname{N}_{\bullet }( [n] \times \operatorname{\mathcal{C}}), \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) ) \\ & \simeq & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{N}_{\bullet }( [n] ) \times \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}), \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) ) \\ & \simeq & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^ n \times \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}), \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) ) \\ & \simeq & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^ n, \operatorname{Fun}( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}), \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) )). \end{eqnarray*}

It will therefore suffice to show that $v$ is bijective, which is a special case of Proposition 1.2.2.1. $\square$